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Analysis of a model for hepatitis C virus transmission that includes the effects of vaccination with waning immunity
TLDR
In this article, a mathematical model based on the transmission dynamics of hepatitis C virus (HCV) infection is considered and the basic reproduction number, $R_0, and the equilibrium solutions of the model are determined.Abstract:
This paper considers a mathematical model based on the transmission dynamics of hepatitis C virus (HCV) infection. In addition to the usual compartments for susceptible, exposed, and infected individuals, this model includes compartments for individuals who are under treatment and those who have had vaccination against HCV infection. It is assumed that the immunity provided by the vaccine fades with time. The basic reproduction number, $R_0$, and the equilibrium solutions of the model are determined. The model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists with a stable endemic equilibrium whenever $R_0$ is less than unity. It is shown that the use of only a perfect vaccine can eliminate backward bifurcation completely. Furthermore, a unique endemic equilibrium of the model is proved to be globally asymptotically stable under certain restrictions on the parameter values. Numerical simulation results are given to support the theoretical predictions. [epidemiological model; equilibrium solutions; backward bifurcation; global asymptotic stability; Lyapunov function.]read more
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Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
TL;DR: A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations and it is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R 0>1,Then it is unstable.
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Backward bifurcations in dengue transmission dynamics
TL;DR: In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence.